9405
J. Phys. Chem. 1994, 98, 9405-9410
FEATURE ARTICLE Adhesion Promoters F. Brochard-Wyart,? P. G. de Gennes,*’sL. LCger,’ Y. Marciano,’ and E. Raphael$ PSI, Institut Curie, 11 rue Pierre et Marie Curie, 75005 Paris, France, and Physique de la Matiere CondensCe, Collkge de France, I1 place M. Berthelot, 75231 Paris Cedex 05, France Received: April 27, 1994@ Adhesion between a flat solid and a rubber is considerably increased by grafting chains (chemically identical to the rubber) on the solid surface. Similarly, a population of mobile chains dissolved into rubber can increase the strength of a rubberhbber contact. A third (more recent) type of promoter is a “Guiselin brush”, obtained by incubating certain polymer melts against a suitable solid. A major practical problem is to understand how the adhesive energy varies with the surface concentration (5 of “connector molecules”: very often, at high connector density, the strength goes down. We set up a theoretical picture of polymer interdigitation, covering most of the cases listed above. We also describe peeling measurements of the adhesion energy G ( o ) for Guiselin brushes (silicaPDMSPDMS network). For the Guiselin brush, the statistical problem is much more complex. The main experimental result is the presence of a clear maximum in the plot of G ( a ) .
I. Various Types of “Connector” Molecules This paper is mainly concerned with the adhesion of rubbers. A first, major example concerns the contact between a rubber and a flat solid. If the solid surface is bare, the adhesion energy G (the energy required to separate a unit area of contact) is very small, comparable to the thermodynamic work of separation, i.e. smaller than 0.1 Jlm2. In practical adhesion problems, we need much larger values of G. This can be achieved by a number of methods. (1) Grafted Chains. A classical approach amounts to attaching terminally certain polymer chains (the “connectors”) on the solid surface. This actually leads to two distinct operation modes: loose bridging, where the connector end wanders into the rubber matrix (Figure la), and chemical bridging, where the connector end carries, for instance, a vinyl group and is ultimately bound to the rubber network (Figure lb). Let us first assume that the number of grafted chains per unit area v (or its dimensionless analog 0 = vaz, where a is a monomer size) is small. Then, the grafted chains do penetrate easily in the rubber, if they are chemically identical to it. The adhesion energy for loose bridging has been analyzed by E. Raphael;’12 the starting point is shown in Figure 2, where we see an advancing fracture with connector chains which are stretched up to a final length hf. When the crack opening exceeds hf, the connectors snap back to the solid surface and dissipate some energy: this is the leading contribution to G (at least in the limit of low fracture velocities; we always stay in this limit). A central observation is that connectors can be pulled out only when the force acting on each of them is beyond a certain t h r e s h o l d p . There are two main contributions t o p : (a) the van der Waals monomerlmonomer attractions U,, disfavor a molecular thread directly exposed to air and (b) the entropy of the thread is reduced. E. Raphael analyzed the combined effects of contributions a and b in a regime where U,, 5 kT. He found that the force$ +
+ + + + + + +
+ + + + + + +
(b)
(a)
Figure 1. Grafted connectors between a solid and a rubber. In our entire discussion, the connector chain is assumed to be chemically identical to the rubber: (a) loose bridge; (b) chemical bridge. /
N
NO
Figure 2. Mobile chain near the interface between two identical rubber blocks. Our entire discussion assumes that a single mobile chain is ideal; this is correct for N < NO*.
is, in fact, a geometric mean between both contributions,’ namely,
PSI,Institut Curie.
* Collbge de France.
e Abstract published in Advance ACS Abstracts,
August 1, 1994.
0022-365419412098-9405$04.5010
For many practical cases, U,, and kT are comparable, and we 0 1994 American Chemical Society
Brochard-Wyart et al.
9406 J. Phys. Chem., Vol. 98, No. 38, 1994
can use the simplified form
f*
-
UvJa
(1’)
-
The other important ingredient is the ultimate length, hf,of the connectors. In the regime kT U,, the chains are nearly fully extended: h, = Nu
(2)
+
+
+
+
+
+
+
Figure 3. Idealized representation of a “pseudobmsh’. A number of
where N is the polymerization index of the connectors. The energy G (in the low-velocity limit) is then simply the work (per unit area) performed to pull out the chains with a forcef nearly equal to the threshold f*:
overlapping chains are bound to the surface, but only one chain has been displayed. The number of contact points is on the order of NIiZ (where N is the number of monomers per chain). The average length of the loops is ii N”z, but some loops are much larger.
G = vf*hf = vNf*a
coil is still given by eq 3! Thus (omitting always all numerical coefficients)
vNUvW= a N y
(3)
where y is the surface tension of a melt of connector molecules. The crucial factor in eq 3 is N: interdigitation leads to large enhancements of adhesion. The remaining problem is the choice of the surface density 0. To obtain strong adhesions, we wish to increase u. But, clearly, when u is very large, we are in danger of losing the interdigitation. The brush may segregate from the network. In section I1 (expanding upon the ideas of ref 3), we shall study what happens at relatively large grafting densities. Let us briefly mention here the case of a brush with chemical bridging to the rubber. Here, we can understand the adhesion energy G following a classical argument by Lake and tho ma^.^ When the connectors are pulled out, each of their bonds has a stored elastic energy which is huge, comparable to a chemical binding energy U,. After scission, the energy dissipated per connector is -NU, and
G
-
(4)
vNUx
Again, this is linear in N! If we compare the two estimates 3 and 4, we find a remarkable law:’ Gchemical GlOCW
I”+ =
ux
-
@RdNa3
(5)
(6)
Or, in dimensionless units
a=(bN- 112
What is the maximum value of (b which can be achieved in practice? This was discussed in ref 3 and will be presented in section 11. It is worth emphasizing that eq 8 is expected to work only in a low-velocity regime. At higher velocities, a completely different process becomes important, namely, the viscous relaxation of the mobile chains, far from the fracture tip. For instance, Ellul and Gent studied rubber blocks mixed with mobile chains of polyi~obutylene~ and found a considerable contribution of this type. (For a simple description of viscoelastic losses in fracture, see ref 8.) (3) Pseudobrushes. When a silica surface is exposed to a melt of poly(dimethylsi1oxane) (PDMS) for a sufficiently long time (-1 day), a certain amount of polymer becomes permanently bound to the solid. This amount is usually measured in terms of an “equivalent dry thickness” h. h is determined by washing out all the unbound polymer by a good solvent and then drying and measuring the residual weight. This has been done for various molecular weights by Cohen Addad and coworkers; they findg h = const N’I2a
vw
Typically, this ratio is on the order of 50. Early experiments by Ahagon and Gent,5 using relatively short silane chains, between glass and a cross-linked p-butadiene, gave ratios on the order of 40. (2) Mobile Chains between Two Rubber Blocks. Figure 2 describes the situation. Each mobile coil has a size Ro = “”a. The volume fraction occupied by the coils inside the rubber is (b. The connectors correspond to coils which are in a sheet of thickness -Ro near the interfacial plane. Their superficial density is thus
v
-
(7)
What is their impact on the adhesion energy G? At first, we note that each coil provides many connecting pieces between the two sides (a simple scaling estimate is N1/*piecedcoil). However, this is misleading. Adhesion with systems of this sort (“many stitch”) was analyzed theoretically by Hong JL6 he showed that, when the crack begins to open, very soon, most of the N112bridges zip out and that the effective strength of the
(9)
where N is always the number of monomerslchain. Equation 9 can be understood as follows: all the coils which touched the silica during incubation were within one radius of gyration of the surface. In this system, each coil is bound to the surface at many points (Figure 3). The surface density of chains is u = h/Na. When the polymer layer is again exposed to a good solvent, it swells; the basic theoretical picture for this form of swollen brush is due to 0. Guiselin.’O Among other things, he found the length distribution of the loops shown in Figure 3. He defines a-%(n) as the number of loops (per unit area), with a number of monomers 2 n. Two conditions define the structure of a(n): (a) The total number of loops (aP2a(1)) is equal to the total number of chains per unit area (a-2u)multiplied by the average number of bridging sitedchains p. A simple argument shows that p (In one coil volume, -Ro3, we have an internal concentration N/RO3;the surface layer is a slice of volume Ro2a.) Thus,
-
where o is always the total number of chains per area u2.
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J. Phys. Chem., Vol. 98, No. 38, 1994 9407
(b) The total number of monomers per unit area is
Equation 17 is nothing more than a simplified version of the Flory theory of swelling, with a particular choice of the reference state. A more precise version would amount to p = kqln 4 NINd] (9 a[,we have a “brush” with extended chains. Here, a[is still given by eq 18! And, by using eqs 16 and 20, our reader can check that the interdigitation is complete only for u < ul. What happens beyond that point? There are many possibilities: (1) One of us (F. Brochard-Wyart) proposed a state of partial interdigitation, where each connector inserts only n( “) monomers into the network. The number n is such that a = ul(n);That is
The resulting adhesion energy is
It is an increasing function of u. ( 2 ) Another type of partial interdigitation corresponds to a fraction f of the chains which penetrate, while a fraction (1 f, does not penetrate at all and forms a passive layer between the solid and the network (Figure 5). If each penetrating chain infiltrates n segments into the rubber, the threshold condition is now
af= o,(n)=
(25)
The thickness h of the passive layer is given by
hlu = o(1 - f , N
+ af(N - n )
(26)
and f must be obtained through a minimization of the free energy F (per unit area). This F contains two contributions: (a) Each of the afpenetrating chains is just at threshold and has an energy -kT due to network swelling. (b) The interface between the passive layer and the network has a certain surface energy, because a number u(l - f, of chains are reflected there. If the interface was sharp, this would
(kT/No1’2a2)( 1 - j)
(27)
and to an overall energy F of the form
Near the onset of partial interdigitation (a
-
q ( N ) ) we have
We constantly assume that N >> NO. Then, for u only slightly smaller than q,the slope dFldf in eq 28 is negative, and the optimum should correspond tof= 1, Le. to the Brochard-Wyart proposal. On the other hand, at higher u values (a the slope would become positive, and the system prefersf= 0. Interdigitation then disappears completely. This prediction is described in Figure 6. It is supported by an analogy: as already pointed out after eq 22, there is (for our purposes) a certain similarity between a network and a liquid of short chains (length NO). The statistics of a brush exposed to such a liquid has been studied in ref 16 and more recently by C. Ligoure17 and M. Aubouy and E. Raphael.’* The limit which all authors find for interdigitation turns out to be exactly the same (a
-
-
IV. Adhesion ElastomeriPseudobrush (1) Experiments. We have prepared series of pseudobrushes formed by simultaneous adsorption and end grafting of a - w hydroxy-terminated poly(dimethylsi1oxane) (PDMS) (narrow fractions, weight average molecular weight in the range 20 OOO700 000) on the naturally oxidized (silica) surface of silicon wafers. The measured dried thickness of these layers obeys the scaling law of eq 13, with u = 5.5 A.12 These PDMS layers were then put into contact with a ribbon of cross-linked PDMS (average molecular weight between crosslinks A40 in the range 10 000-70 000) for at least 2 days. After this time, the adhesive strength of the interface elastomer/ pseudobrush was tested, using a 90” peel test conducted at very low velocity (50-500 h). The details of the experimental setup will be reported e 1 s e ~ h e r e . l Typical ~ results for the energy (per unit area) necessary to break the junction at a velocity 50 &s are reported in Figure 7 as a function of 40,for a series of pseudobrushes formed with PDMS of molecular
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J. Phys. Chem., Vol. 98, No. 38, 1994 9409 this argument, one would naively expect a rather weak adhesion G(a): the number 6of loops of length L m which are allowed to enter is now
1.5
(1.
h
N
E
5
v
0 = am-’’’
1
rather than a(Nlm)1/2. The adhesion energy should then be given by
Gl2y G fNoN dm(am-3‘2)mE ON1/2
ao
0’5
.1€
+ I
t
??
(rather than ON for a simple brush). Equation 32 would hold for full interdigitation. Ultimately, at high a, eq 32 would be cut off, possibly again at u = as in simple brushes. The adhesion energies measured in the PDMS experiments appear somewhat too high to be described by eq 32.
V. Concluding Remarks
0
0
(31)
0.1
0.3
0.2
0.4
0.7
0.6
0.5
0.8 0.9
1
Figure 7. Experimental results for the adhesion energy between pseudobrushes formed from PDMS of molecular weight 242 370 and a PDMS elastomer (molecular weight bFtween cross-links M,= 10 250 measured through 90’ peel tests at 50 Ah,for series of pseudobrushes formed at various volume fractions of polymer in the reaction bath @O.
t
t
t
t
t- t
I
t
J
+
t
The “cactus”. One loop of a pseudobru:-- (IJ), when Figure penetrating a network, builds up a conformation where every “gate” (for instance the gS gate) is used at least twice; this imposes a strong reduction in entropy.
weight 242 370 ( I = Mw/Mn= 1.10) and an elastomer with MO = 10 250. The most remarkable feature is the existence of a maximum in the adhesive strength of the elastomerlpseudobrush interface, when 40is increased. This is in good qualitative agreement with what has just been predicted in section I11 for the brushes, with an interdigitation, and thus an adhesion energy vanishing at high u. A pseudobrush is, however, much more complicated to describe than a brush, and we do not have yet a theory for this case. (2) Theoretical Aspects. A pseudobrush is not really equivalent to a polydisperse brush. A major feature is the “cactus effect” displayed in Figure 8. When a loop with m monomers and fixed attachment points (I and J) attempts to interdigitate with the network, its conformations are strongly restricted, and there is an entropy loss:
where p is a numerical factor of order unity. The net result is that most loops with m > NOshould not enter at all. The only pieces which enter freely are the two ends of the chain. From
At the crude level of scaling laws, we begin to have a certain perception of the main controlling factors for rubber adhesion in the presence of adhesion promoters, or “connector molecules”. (1) For rubbedrubber contacts, strengthened by mobile chains, the history of the rubber is crucial: if it has been cross-linked prior to the insertion of mobile chains, the maximum concentration of connectors should be severely limited, and short connectors are preferable, as seen from eqs 3 and 18. (2) For simple grafted layers, we predict a regime of partial interdigitation, where the adhesion energy increases very slowly with the grafting density. At higher surface concentrations (a > No-l/z),the brush should not interdigitate any more. All this should be amenable to experiments, possibly by using block copolymers rather than grafted chains. (3) With a typical model system such as PDMS against silica, the most readily available situation corresponds to the Guiselin brush. Our peeling data on these brushes show an interesting maximum in G(o).However, no complete theory is available yet for this case. (4) There are other limitations to our discussion, such as the following: (a) Rubberlrubber contacts (or latextlatex contacts) will be, in practice, extremely sensitive to the polydispersity of the mobile connectors. This can, in fact, be taken into account rather easily in our picture. (b) We have not incorporated the possibility of chemical scission for the connectors. In fact, our discussion of section I shows that, for most loose systems, the force on the connectors (at low velocity) is$ = U,,/a, where U,, is a van der Waals energy. This is far below the chemical rupture forcesf, U,/ a. Thus, in the low-velocity regime, scission is indeed negligible for loose systems of unbranched connectors.
-
Acknowledgment. We have benefited from very stimulating exchanges with M. Aubouy, J. C. Daniel, M. Deruelle, H. Hervet, C. Ligoure, H. Brown, G. Schorsch, and P. Silberzan. The experimental part of this work has benefited from support by the RhBne-Poulenc Co. References and Notes (1) Raphael, E.; de Gennes, P. G. J . Phys. Chem. 1992, 96, 4002. (2) Brown, H.R.; Hui, C. Y.; Raphael, E. Macromolecules 1994, 27, 608. (3) de Gennes, P. G. C. R . Acad. Sci. Paris 1994, 318 (II), 165. (4) Lake, G. J.; Thomas, A. G. Proc. R . SOC.London, A 1967, 300, 108. (5) Ahagon, A.; Gent, A. J . Polym. Sci., Polym. Phys. Ed. 1975, 13, 1285.
9410 J. Phys. Chem., Vol. 98, No. 38, 1994 (6) Ji, H.; de Gennes, P. G. Macromolecules 1993, 26, 520. (7) ElIuI, M. D.; Gent, A. J. Polym. Sci., Polym. Phys. Ed. 1984, 22, 1953. (8) de Gennes, P. G. C. R . Acad. Sci. Paris 1988, 307, 11. (9) Cohen Addad, J. P.; Viallat, A. M.; Pouchelon, A. Polymer 1986, 27, 843. (10) Guiselin, 0. Europhys. Left. 1992, 17, 225. (11) Auvray, L.; Auroy, P.; Cruz, M. J. Phys. (Paris) 1992,2 (6), 943, (12) Deruelle, M.; Ltger, L. To be published.
Brochard-Wyart et al. (13) This law was first described by M. Daoud. See for instance: P.
G.de Gennes, Scaling laws in polymer physics; Cornell Univ. Press, 2nd printing, 1985; Chapter III. (14) (15) (16) (17) (18) (19)
Bastide, J.; Candau, S.; Leibler, L. Macromolecules 1980,14,719. Brochard, F. J . Phys. (Paris) 1981, 42, 505. de Gennes, P. G . Macromolecules 1980, 13, 1069. Ligoure, C. Thbse Univ. Paris 6, 1991. Aubouy, M.; Raphael, E. Macromolecules, to be published. Marciano, Y.;Ltger, L. To be published.
